Algebraic structure of the manifold of solutions of the $N$-body problem

Abstract

The Theorem of Ritt on the decomposition of the perfect differential ideal generated by a single irreducible differential polynomial is, here, generalized to system of polynomials satisfying certain conditions. We use these results to prove that all solutions of the N-body problem, excepting the solutions for which one or more of the r,; (the distance between the masses Mi, Mj) is zero, belong to one irreducible manifold. Introduction. In [3, p. 22] Ritt proved that the manifold of zero's of a system of differential polynomials is a finite union of irreducible manifolds. If the system consists of a single irreducible differential polynomial F, then the manifold of F breaks up into an irreducible general manifold and a certain number, possibly zero, of singular manifolds. In this paper the latter result is generalized to systems of polynomials: F= F1, * * * , F., n > 1 satisfying certain conditions. These conditions are almost always satisfied in problems of particle mechanics. When we apply these results to the N-body problem, we prove that all solutions of the N-body problem, excepting those for which one or more of the rij (the distance between the point masses Mj, Mj) is zero, belong to one irreducible manifold (the general manifold). The solutions of the N-body problem for which one or more of the rij is zero constitute the singular manifolds. In particular, the equilateral triangle, isosceles triangle, straight line solutions of the three-body problem belong to the general manifold. However, this paper still does not answer the question whether any singular solution belongs to the general manifold. For the applications to mechanics only Theorem 1 and Corollary 1 is needed and the number of variables can be assumed to be the same as the number of equations. In order to obtain a proper generalization of Ritt's Theorem on the decomposition of a single differential polynomial in an arbitrary number of variables we have proven our results for the case where the number of variables is > the number of equations, and extended these results to systems of greater generality than that required for problems in particle mechanics. Received by the editors July 23, 1969. AMS Subject Classifications. Primary 1280, 8520.